# 5.3 UPSCALING: BRIDGING THE ATOMIC LEVEL TO THE DISLOCATION DENSITY LEVEL AND THE CONTINUUM LEVEL

There are several upscaling bridges from the atomistic level to the continuum and dislocation dynamics levels related to plasticity, fracture, and fatigue. Also relevant are the notions of stress and strain at the atomic level. What makes these continuum quantities interesting is that they are defined not as discrete quantities but as field variables. We next describe the different quantities that are used for upscaling.

## 5.3.1 Continuum Quantities for Upscaling

### 5.3.1.1 Atomic Stress.

As mentioned, the notion of a stress tensor is a continuum quantity, so determining the values of the components of the stress tensor and its associated quantities relevant to upscaling such as the invariants, effective stress, and stress triaxiality might at first blush seem not possible to calculate. However, the most intriguing definition of stress related to modern multiscale modeling is the virial theorem stress that was first discovered by Clausius in 1870, and further developed by Maxwell, Rayleigh, Poincare, and Eddington. The virial theorem is a useful theoretical tool for understanding some of the collective behaviors of a large number of atoms or particles. It is widely used for various interpretations of modern MD simulations. Its application scope includes classical mechanics, quantum mechanics, and relativity mechanics.

Maxwell discovered the virial theorem almost simultaneously with Clausius. ...